Dendrites help mitigate the plasticity-stability dilemma

With Hebbian learning ‘who fires together wires together’, well-known problems arise. Hebbian plasticity can cause unstable network dynamics and overwrite stored memories. Because the known homeostatic plasticity mechanisms tend to be too slow to combat unstable dynamics, it has been proposed that plasticity must be highly gated and synaptic strengths limited. While solving the issue of stability, gating and limiting plasticity does not solve the stability-plasticity dilemma. We propose that dendrites enable both stable network dynamics and considerable synaptic changes, as they allow the gating of plasticity in a compartment-specific manner. We investigate how gating plasticity influences network stability in plastic balanced spiking networks of neurons with dendrites. We compare how different ways to gate plasticity, namely via modulating excitability, learning rate, and inhibition increase stability. We investigate how dendritic versus perisomatic gating allows for different amounts of weight changes in stable networks. We suggest that the compartmentalisation of pyramidal cells enables dendritic synaptic changes while maintaining stability. We show that the coupling between dendrite and soma is critical for the plasticity-stability trade-off. Finally, we show that spatially restricted plasticity additionally improves stability.


Inhibitory Plasticity
Synapses from inhibitory neuron j to excitatory neuron i change their weight w I ij according to Vogels et al. (2011) dw I ij dt =η I z I j (t)S i (t) where η I is the inhibitory learning rate, z I j (t)/z I i (t) is the pre/post-synaptic trace, S j (t)/S i (t) is the pre/post-synaptic spike train, and α I = 2κτ iST DP determines the amount of transmitter-induced depression (Vogels et al. 2013). κ is the target firing rate, which was determined by the average population firing rate of the last 2 s of the warm-up phase. Plastic inhibitory weights are limited by a maximum weight w I max = 100nS. The pre/post-synaptic traces z I j (t)/z I i (t) are written as   Tables in the  main manuscript), if not mentioned otherwise in the following: For the network without dendrites, we used the same parameters, except that g s was set to 0 and we placed the excitatory and inhibitory synapses that used to be on the dendrite in the original network (w d EE and w d EI ) onto the soma. For the simulations with inhibitory plasticity, we added inhibitory plasticity (see section "Inhibitory Plasticity") only to those synapses that are on the dendrite (w d EI ) in the model with dendrites, or to the fraction of additional synapses (that used to be on the dendrite) on the soma in the model without dendrites. In all simulations, the learning rate η of excitatory plasticity was doubled to 10.
Gating plasticity in two-compartment versus single-compartment networks For Suppl. Fig. S5, we simulated a network of the same size, where excitatory cells were single-compartment neurons. The somatic membrane equation was the same as for the two-compartment neurons with g s = 0 (no coupling to the dendrite). All parameters were the same as in the original network, with two exceptions, as we placed all synapses on the perisomatic compartment. First, the connection probability from excitatory to excitatory cells was increased to p EE = 0.19 to account for the additional synapses that were previously placed on the dendrite. Second, in addition to the inhibitory synapses the soma already contained in the original simulation, it contained inhibitory synapses with the connection strength of inhibitory synapses on the dendrite in the original model w d EI = 4.0nS with a probability of 0.1.

Network with inhibitory subtypes
For Suppl. Fig. S6, we included two populations of inhibitory celltypes, corresponding to parvalbumin-positive (PV) and somatostatin-positive (SST) interneurons. The PVs exclusively target the soma of the excitatory cells, and the SSTs exclusively target the dendrites of the excitatory cells. Excitatory cells and PVs, but not SSTs received feedforward Poisson inputs.

Parameter
Value

NMDA conductances
In Suppl. Fig. S7, the excitatory conductance g E for each neuron i is composed of an AMPA and a an NMDA term as in Zenke et al. (2013).

Synaptic Scaling
In Suppl. Fig. S9, instead of a sliding depression amplitude (Bienenstock et al., 1982), we used synaptic scaling as a homeostatic mechanism, implemented by the term γ(κ −s i ) in the following weight update equations. As befores i is the moving average of neuron i's activity, κ is the target firing rate. For perisomatic synapses, weights change according to: where Equivalently, for the dendritic synapses, weights change according to: A − i was fixed to 2.7e-3.

Memory network
For Suppl. Fig. 10 we used the described network and added plasticity on inhibitory to excitatory synapses (see section "Inhibitory plasticity") and a competition mechanism for postsynaptic weights.
Heterosynaptic depression If the sum of postsynaptic weights in the perisomatic compartment or the dendrite exceeds a maximum 1.5pN E w 0 (hard bound), all perisomatic/dendritic synaptic weights are scaled down equally by the average synaptic weight change of the postsynaptic compartment in the current time step, that is the total perisomatic/dendritic weight change divided by the number of incoming perisomatic/dendritic synapses.
Stimulation protocol We first simulated the non-plastic network for 5 s to calculate the steady state population firing rate. We then introduced plasticity with a target firing rate κ of the measured population firing rate and simulated the network with plasticity for another 5 s. Then, we activated pattern P1 for 3 s, which was realised by an external input of 100 Poisson spike trains with a firing rate of 20 Hz to neurons with indices 400 to 499 (ensemble E1). Afterwards, we changed the gating variable under investigation (except in Fig. 10a, where we did not apply any gating). After a stimulation pause of 7 s, we activated pattern P2 for 5 s, which was realised by an external input of 100 Poisson spike trains with a firing rate of 30 Hz to neurons with indices 450 to 549 (ensemble E2). We continued the simulation for further 10 s without stimulation.

Supplementary Figures
Dendritic weight changes as a function of τ crit in the network simulations shown in Fig  To make the two models comparable, we add the dendritic (excitatory and inhibitory) synapses to the soma in the soma only model with the same connection probability and strength as in the model with dendrites. Then we increase the learning rate by a factor of two of those excitatory synapses on the dendrite (in the model with dendrites) and of those excitatory synapses that used to be on the dendrite (in the model without dendrites). Similarly, we add inhibitory plasticity (see Suppl. Methods) to the inhibitory synapses on the dendrites (in the model with dendrites) and to the inhibitory synapses on the soma that used to be on the dendrite (in the model without dendrites). Inhibitory plasticity in the model with dendrites (a) increases both the tolerated critical time constant (compare to c without inhibitory plasticity), and it allows for more synaptic change in the dendrite (b, compare to d). In the model without dendrites however, the stability does not change very much with the same amount of inhibitory plasticity (compare only soma in a and c) and the synaptic weight changes are remain small (compare only soma in b and d).
Effect of the dendritic nonlinearity. Comparison of gating plasticity in the dendrites in a network with two-compartment neurons to gating plasticity in the perisomatic compartment in a network with single-compartment neurons.  The network with synaptic scaling as a homeostatic mechanism.
S 9. The network with synaptic scaling as a homeostatic mechanism. Raster plots of a network with synaptic scaling as a homeostatic mechanism. a: no gate applied, synaptic scaling with a homeostatic time constant of 10s. b-e: synaptic scaling with a homeostatic time constant of 50s. b: no gate applied. c: reduced learning rate (reduced by 40%). d: reduced excitability (reduced by 10%). d: increased inhibition (increased by 10%).
Plasticity-inducing stimulation of the network and protection of memories by gating plasticity.
In addition to the impact on the stability of network activity, plasticity interferes with the stability of memories. In a plastic network, neural activity patterns lead to synaptic changes, which could overwrite memories that were previously stored in the synaptic connections. Especially when resources are limited, forming new memories can come at the expense of old ones. By gating plasticity, memories can in principle be protected. For example, trivially, by switching off plasticity, old memories are protected. Interestingly, the gates we considered here are not simple all-or-nothing gates but can be continuously modulated. We, therefore, investigated how their modulation affects the maintenance and encoding of memories. Note that we do not use spatially localised gating here. Before gating plasticity, we first tested how a memory interferes with a previously stored memory in a plastic network. We added inhibitory plasticity (Vogels et al., 2011) to ensure network stability during memory formation and heterosynaptic depression to introduce synaptic competition (see Suppl. Methods). First, we showed pattern P1 to the network (Fig.  S10a). That is, a subpopulation of 100 excitatory neurons received excitatory Poisson input (100 Poisson spike trains with a firing rate of 20 Hz). Afterwards, neurons activated by P1 formed a neural ensemble E1 by increasing their connectivity (Fig. S10a top left). Then, we showed a pattern P2 to the network. P2 is similar to P1 and hence activated a group of neurons E2 that overlapped with the previously formed ensemble. Neurons activated by P2 increased their connectivity. Because the patterns overlapped, synapses were increased at the expense of connections from the old memory (Fig. S10a  bottom). Therefore, the new memory interfered with the old memory (Fig. S10 top right). We defined the difference between the mean connection strength of the P1 neurons after memory formation of P1 and the mean connection strength of the P1 neurons at the end of the simulation -after P2 has been learned -as the breakdown of the memory.
To test the effect of gating on the protection of memories, we applied the different gates after the first memory is formed. Reducing learning rate to 0 after the first memory is formed trivially protects the memory from being overwritten by the second pattern (Fig. S10b), as this blocked further weight changes. It is less clear how a change in excitability or inhibition affect the storage of the memory. Unlike the learning rate, we cannot modulate excitability or inhibition to their extremes without silencing neural activity. On the one hand, because they decrease neural firing rates, these gates could protect memories by reducing weight changes. On the other hand, by decreasing firing rates, they could also increase LTD as experimentally, low firing rates promote more LTD than LTP (Dudek and Bear, 1992; Sjöström et al. 2001).
We found that a reduction in excitability could indeed protect the memory (Fig. S10c) without permanently silencing the network (Fig. S11c). With lower excitability, the stimulated neural ensemble, E2, fired at a lower rate (compare E2 in Figs. S11g and S11e), and weights within E2, including those projecting to the overlapping ensemble, O, potentiated less (compare Fig. S10c bottom with Fig. S10a bottom). The new ensemble, E2, hence competed to a lesser extent with the old memory, leading to less memory decay due to heterosynaptic depression. Note that protecting the old memory hence comes at the cost of storing an equally sized representation of P2.
Similar to reduced excitability, increased inhibition could also protect the memory (Fig. S10c) as it reduced firing rates in the network (Fig. S9d,h). Notably, the inhibitory plasticity in the network additionally protected the previously formed memory, as it led to increased inhibition of the old memory ensemble E1 (Fig. S11i). This further reduced potentiation of synapses from the new ensemble, E2, to the overlapping ensemble, O.
Because decreased excitability and increased inhibition lower the firing rate of the network, we asked whether this low firing rate induces LTD and hence counteracts the protection of the old memory ensemble. We found that there was no increased LTD within the old memory ensemble due to low firing rates. First, the non-overlapping population of the old ensemble fired at a very low rate (E1-O in Fig. S11g,h). There was hence little depression from the non-overlapping population to the overlapping one, as depression happens upon presynaptic spiking. Second, the memory breakdown was weaker at lower excitability and higher inhibition, i.e. at lower firing rates (Fig. S10f,g). The old memory was hence mostly at risk due to heterosynaptic depression. In line with this, the memory breakdown correlated with the maximum mean strength of synaptic connections to the overlapping ensemble O during pattern P2 (Fig. S11j-l).
The specificity of the protective effect depends on which gating mechanism is used. For the learning rate, the effect is specific at the synaptic level. For excitability and inhibition, the effect is specific at the cell level. Excitability, inhibition, and learning rate can be modulated at the compartmental level and introduce further specificity there.
When we continuously modulated the gates, we found that the breakdown of the memory increased nonlinearly with both increasing excitability (Fig. S10f) and decreasing inhibition (Fig. S10g), and increased linearly with learning rate (Fig. S10e).
To conclude, all gates can protect memories. Learning rate can be modulated independent of network activity and hence act as a switch for plasticity. Although excitability and inhibition do not modulate plasticity separately, they can both protect memories by reducing activity and weight changes without silencing network activity. S 10. Plasticity-inducing stimulation of the network and protection of memories by gating plasticity. a-d: After 10 seconds, we show pattern 1 (P1, grey) to the network for 3 seconds. After a gap of 7 seconds, we show pattern 2 (P2, green) to the network for 5 seconds. Top panels: excitatory weight matrix at two time points of the simulation. Bottom panel: mean synaptic weight from ensemble 1 (E1) to the overlapping region (grey) and from ensemble 2 (E2) to the overlapping region over time as an indication of the strength of the memory of P1 and P2, respectively. a: no gating. b: after P1 is learned, the learning rate is set to 0 (denoted by purple background). c: after P1 is learned, excitability is reduced by 50% (denoted by blue background). d: after P1 is learned, inhibition is doubled (denoted by green background). e-g: breakdown of the memory (see a) as a function of learning rate (e), excitability (f), and inhibition (g).